Perturbation analysis of third-order tensor eigenvalue problem based on tensor-tensor multiplication
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Perturbation analysis has been primarily considered to be one of the main issues in many fields and considerable progress, especially getting involved with matrices, has been made from then to now. In this paper, we pay our attention to the perturbation analysis subject on tensor eigenvalues under tensor-tensor multiplication sense; and also ϵ-pseudospectra theory for third-order tensors. The definition of the generalized T-eigenvalue of third-order tensors is given. Several classical results, such as the Bauer-Fike theorem and its general case, Gershgorin circle theorem and Kahan theorem, are extended from matrix to tensor case. The study on ϵ-pseudospectra of tensors is presented, together with various pseudospectra properties and numerical examples which show the boundaries of the ϵ-pseudospectra of certain tensors under different levels.
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